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On the classification of $2$-distance-primitive graphs of small valency

  • Xing Zhang
  • Xue Wang
  • Jun-Jie Huang
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Abstract

A non-complete vertex-transitive graph is called 2-distance-primitive if a vertex stabilizer acts primitively on both the first and second distance neighborhoods of the vertex. Recently, Ruan, Liu and Jin [Discrete Math. 347 (2024), 113978] showed that every connected 2-distance-primitive graph of valency at most 9 either belongs to certain known families or satisfies a collection of further restrictions, with the socle of a vertex stabilizer isomorphic to PSL(2,7) or PSL(2,8). In this paper, we resolve these remaining cases and thereby obtain a complete classification of connected 2-distance-primitive graphs of valency at most 9. We also give a short proof of the main result of Ruan, Liu and Jin and identify additional graphs omitted from their list.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026classification,
  title = {{On the classification of $2$-distance-primitive graphs of small valency}},
  author = {Xing Zhang and Xue Wang and Jun-Jie Huang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07733},
  eprint = {2611.07733},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07733},
}
RIS
TY  - UNPB
TI  - On the classification of $2$-distance-primitive graphs of small valency
AU  - Xing Zhang
AU  - Xue Wang
AU  - Jun-Jie Huang
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07733
UR  - https://arxiv.org/abs/2611.07733
AB  - A non-complete vertex-transitive graph is called 2-distance-primitive if a vertex stabilizer acts primitively on both the first and second distance neighborhoods of the vertex. Recently, Ruan, Liu and Jin [Discrete Math. 347 (2024), 113978] showed that every connected 2-distance-primitive graph of valency at most 9 either belongs to certain known families or satisfies a collection of further restrictions, with the socle of a vertex stabilizer isomorphic to PSL(2,7) or PSL(2,8). In this paper, we resolve these remaining cases and thereby obtain a complete classification of connected 2-distance-primitive graphs of valency at most 9. We also give a short proof of the main result of Ruan, Liu and Jin and identify additional graphs omitted from their list.
ER  -